Quote dairymaid: I believe he loves Katherine but does not trust her. Plus, in the Originals pilot episode (near the end) he kissed her on the forehead and said goodbye. So, I do not know where the relationship will go from here or if Katherine will go to NOLA to try to get him to accept her back in his life or not.

She did seem surprise that he just walked away...... :)

Elijah doesn't like to be lied to!!!! And she did exactly that! She can do it to others but not to him! Or cos Klaus wouldn't let go of his grudge and he chose to stand by his brother! ;)